Definition:Structure (Set Theory)

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Let $A$ be a class.

Let $\RR$ be a relation.

The relational structure $\struct {A, \RR}$ satisfies well-formed formula $p$, denoted $\struct {A, \RR} \models p$, shall be defined on the well-formed parts of $p$:

\(\ds \struct {A, \RR} \models x \in y\) \(\iff\) \(\ds \paren {x \in A \land y \in A \land x \mathrel \RR y}\)
\(\ds \struct {A, \RR} \models \neg p\) \(\iff\) \(\ds \neg \struct {A, \RR} \models p\)
\(\ds \struct {A, \RR} \models \paren {p \land q}\) \(\iff\) \(\ds \paren {\struct {A, \RR} \models p \land \struct {A, \RR} \models q}\)
\(\ds \struct {A, \RR} \models \forall x: \map P x\) \(\iff\) \(\ds \forall x \in A: \struct {A, \RR} \models \map P x\)